What Did Glaucon Draw?A Diagrammatic Proof for Plato's Divided Line
Despite the frequency and intensity of scholarly attention to Plato's important divided line passage in Republic 6, an ancient puzzle has repeatedly baffled interpreters: the two middle segments of the line are of equal length, although Socrates declares that the degree of clarity and truth increases in proportion to the contents of each of the line's four segments. The missing piece of the puzzle, I argue, is provided by the practice of ancient Greek geometry: diagrams were drawn with a compass and straightedge. Thus, as actually constructed in the fifth and fourth centuries BCE, the diagram itself vindicates Plato's text.
Plato, Republic, divided line, mathematics, geometry, diagram, proof
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1. the division and the mystery
Elaborating the analogy between the sun and the good, Plato's Socrates tells Glaucon to divide a line αβ into two unequal segments at γ. The result is that αγ represents what is intelligible and γβ what is visible.1 Then Glaucon is to divide each of the two segments by the same ratio as he used in the original division (Republic 509d6–8).2 Whatever proportion he used to make the cuts γ, δ, and ε in the divided line, generating its four segments, the geometrical implication is that the two middle segments must be equal in length. As both Nicholas D. Smith and Richard Foley have emphasized, when Socrates reiterates the characteristics of the line at 534a3–5, transposing δγ and γε, there should be no doubt that Plato knew the two middle segments were equal.3 Their equality constitutes half the mystery of the divided line that my interpretation attempts to solve. But let us follow the text awhile.
"[D]ifferences in relative clarity and obscurity" (d9)4 determine what is assigned to each part of the line, its contents, now labeled with uppercase letters. The lower part of what is visible, Δ, represents what is most obscure: images, shadows and reflections; the next part, Γ, represents spatiotemporal objects, sensible particulars, such as animals, plants, and artifacts. The lower part of what is intelligible, Β, represents hypotheses such as those used in geometry and arithmetic; and the upper intelligible part, Α, represents "what reason itself grasps through the power of dialectic" (511b3), namely, Platonic forms. Corresponding to each sort of content is a cognitive state (511d8–e1): A for the contents of highest segment, αδ, νόησις or ἐπιστήμη; Β for δγ, διάνοια; Γ for γε, πίστις; and Δ for εβ, εἰκασία, the lowest of the four segments. [End Page 2]
After Glaucon shows that he has understood Socrates's description of the parts of the line, their contents, and their relations of dependence on one another—that each is the image of the part above it—Socrates says, "arrange them in proportion to their clarity, taking degree of clarity as corresponding to the degree that the things they are assigned to5 share in truth" (511e2–4).
If clarity and truth increase in proportion to the contents of each part of the divided line, as Socrates claims they do, then one would expect the degree of increase to be visible in the diagram. Therein lies the other half of the mystery: there is an apparent contradiction between the initial instructions for dividing the line (509d6–8) and the later description of the line's divisions (511e2–4).6 In short, then, if the two middle segments of the divided line are, and must be, equal, in conformity with Socrates's explicit instructions, then the illustration fails to exhibit the greater degree of clarity and truth as one moves up the line. On the other hand, (a) if the two middle segments contain the very same things, even if used differently;7 or (b) if the stuff of an object just is given by its formal characteristics, if its formal structure and its matter are two aspects of the same one thing,8 then B = Γ without remainder, and the claim of a greater degree of clarity and truth is, at the very least, seriously misleading. Either way, the mystery is evident in all previous representations of Plato's divided line.
Foley helpfully discusses four types of "imperfect solutions" to the contradiction, which he dubs the "overdetermination problem." (i) Early revisionist philologists challenged the text of Republic, suggesting emendations to edit away the contradiction. (ii) Those attempting demarcation denied that the two middle segments of the line were ever intended by Plato to be compared. (iii) A third group argued that Plato made a gaffe in his design of the line, even if he did recognize the equality of B and Γ. My interpretation will rule out those first three approaches. (iv) Finally, dissolution scholars variously emphasized the equality of B and Γ, arguing that Plato wanted especially to draw attention to it.9 Some of the fourth group denied the existence of mathematical intermediaries, a topic with a vast literature [End Page 3] of its own; for Foley's purposes, it is more salient to point out that this fourth imperfect solution disarmed the ontological version of the overdetermination problem—equality of the middle segments. The epistemic version remained because, although διάνοια must be clearer than πίστις, the line gives them equal clarity. Foley credits Smith with being the only philosopher previously to recognize the remaining epistemic problem with the "dissolution" account. I acknowledge their insight that existing representations of the divided line give rise to both aspects of the overdetermination problem, and that any diagram worth its salt must address both satisfactorily, but I stop short of accepting Smith's or Foley's conclusions, though I am sympathetic to how they reached them.10
Growing up with algebra, and knowing in advance what one expects of Plato's divided line, one can easily produce the proportions that have the appearance required by Socrates's initial instructions to Glaucon at 509d6–8. In drawing the diagram above, the desired ratios
αγ : γβ :: αδ : δγ :: γε : εβ
are achieved effortlessly by thinking in units: 6 : 3 :: 4 : 2 :: 2 : 1.11 Getting the ratios right, however, regardless of the length of the units, yields a contradiction with the desiderata of 511e2–4 that has bedeviled discussions of the divided line:
(δγ = γε) & (δγ > γε)
For us, it is straightforward, quick—not to mention pedagogically attractive—to illustrate the line as 4–3–2–1: four units for forms, three for hypotheticals, two for objects, and one unit for shadows. But it would be wrong.12
The divided line passages present a challenge in the interpretation of Plato's philosophy in that, on the one hand, the divided line itself can be seen as a mere analogy to illustrate the relationship between ontological and epistemological concerns, yet on the other—given the crucial importance of mathematics for one's understanding of these relations in this part of the Republic 13—it should strike one as odd that the effectiveness of the analogy should break down as soon as one attempts to produce a diagram of its relations. For it calls immediately [End Page 4] into question Socrates's or Plato's own grasp of elementary mathematics because what calls for a straightforward correspondence between increasing clarity and increasing quantity is complicated by ratios that fail to produce the required image.
I will demonstrate that this pressing difficulty is naturally overcome by shifting from a simple linear representation to the more complex two-dimensional figure that would already have been visible in Glaucon's drawing as a result of the constructions required for making cuts δ and ε. That is to say, I allow the mathematical implications to be the driving force behind the illustration of the divided line: the desired proportions must be visible and the two middle segments of the line must be and appear to be equal. The result of this approach, if I am right, is that the apparent contradiction that Foley rightly identifies as having "exerted a powerful influence on the development of interpretations of the divided line" is resolved both ontologically and epistemically.14
Plato's theory of images obviates the expectation that any image could be perfect, but two advantages of reproducing Glaucon's actual drawing, as I hope to show, are that philosophers no longer need be distracted by a non-problem, and that a whole range of previous emendations and interpretations of Plato's text are ruled out. The remaining difficulties are challenging enough, even after the visual aid to the passage has been restored.
2. stepping stones
The solution I offer below, if I am right, overturns or problematizes from a different angle most of the solutions surveyed by Smith and Foley. However, because my solution is based on the practice of ancient Greek geometry, other relevant precedents—archaeological as well as mathematical—come into play.15
Before the middle of the sixth century BCE, architectural drawings were exact and materials were already available: not just drawing with a stick in the sand or on wax-covered wooden slates, but precisely planed marble slabs used in the building trades.16 And mathematics too, including the use of diagrams as proofs,17 had reached a high level of sophistication long before Socrates was born. Mathematics was taught in Athenian schools, as the geometry lesson in Plato's Theaetetus illustrates, and as the Republic curriculum, inter alia, remind us. How, [End Page 5] then, could twentieth-century philologists and philosophers have surmised that Plato did not have sufficient mathematical understanding to realize that the two middle segments of the divided line were equal?
The answer is reported by historians of mathematics who point to a nadir in the appreciation of geometrical proofs in which Plato was caught up:18 as nineteenth-century mathematicians explored the limits of infinite processes that defied the visual imagination, suspicion of geometrical intuition took hold. That "visual understanding actually conflicts with the truths of analysis" became dogma in the early twentieth century.19 Plato and his serious use of diagrams was only one of the casualties of the dogma. Athenian mathematical understanding generally was disparaged for its reliance on geometry; and mathematical discoveries were pushed from the sixth to the fourth century BCE, eclipsing nearly two centuries. For example, the discovery of the phenomenon of incommensurability came to be viewed as a crisis of Plato's era; in fact, by the time of Socrates, incommensurability was already providing opportunities for significant mathematical work, and much of it involved construction.20 Philosophers were slower than historians of mathematics to reestablish what Athenians knew and when. Key to my argument is that the process of constructing figures was essential to proofs. The intermediate steps in constructing a line divided in the given proportions reveal connections not apparent in mere algebraic manipulation of symbols in the modern formal sense.
An excellent example of diagrammatic proof is nested in the divided line. A Greek schoolboy in Socrates's time could have proved the two middle segments of the divided line must be equal, but almost certainly not with Jacob Klein's proof "in the Greek manner," based on the fifth book of Euclid's Elements:21
Let there be given a line subdivided into four sections.
Let these sections be designated by the letters A, B, Γ, Δ respectively.
Let the division be made according to the prescription: (A + B) : (Γ + Δ) :: A : B :: Γ : Δ.
From (A + B) : (Γ + Δ) :: Γ : Δ follows alternando (Euclid 5.16)  (A + B) : Γ :: (Γ + Δ) : Δ.
From A:B :: Γ : D follows componendo (Euclid 5.18)  (A + B) : B :: (Γ + Δ) : Δ.
Therefore (Euclid 5.11)  (A + B) : Γ :: (A + B) : B. and consequently (Euclid 5.9)  Γ = B.
There is nothing wrong with the proof, and Elements 5 is very likely the work of Eudoxus (±390–340)22 but, in section 4 below, I offer a Euclidean proof based on Elements 1 that could have been used much earlier, and which reaches the same conclusion. [End Page 6]
So far as I have been able to find, we owe the earliest modern attempt at a rigorous description of the proportions of Plato's divided line to the scientific polymath, William Whewell. On November 10, 1856, this former Cambridge second Wrangler addressed the Cambridge Philosophical Society on the subject of Plato's "diagram by which he illustrates the different degrees of knowledge."23 After citing and translating the Greek, Whewell summarizes, "The four segments might be as 4 : 2 :: 2 : 1; or as 9 : 6 :: 6 : 4; or generally, as a : ar :: ar : ar2."24 Nicholas Rescher takes up what he calls the "Whewell Relations" to urge a shift from seeing Plato's line as an analogy to a robust appreciation of its mathematical proportionalities, and what they imply for human reasoning and knowledge.25
A later stalwart, Robert S. Brumbaugh, notably unwilling to lay ignorance or a blunder at Plato's feet, and equally unwilling to assign blame to his copyists, rightly points out what Plato acknowledges himself: images are imperfect. He pays respect to a Plato who has struggled with the illustration of philosophically significant ontological and epistemological material. In his earliest work on the line, Brumbaugh identified the cause of the overdetermination problem (though he did not call it that) as the "interference of metaphor," in which two lines of thought require two different concrete images and neither is appropriate for the other: "if, for example, four things in a set are unequal in respect to property P, yet tightly connected by property Q, a linear representation of these four things as segments cannot adequately present P and Q as simultaneous principles of order." In other words, there is a conflict between the metaphor of proportion and that of disparity and, thus, someone making the diagram "must then choose which of the two properties, P or Q, he intends to make central and which peripheral to his illustration."26 He concludes, "In short, to schematize the passage adequately, the reader should have two distinct diagrams, and be mistrustful of both."27 Brumbaugh was right about two issues crucial to what Glaucon drew: his diagram could not have been linear, and it must have enabled the observer to distinguish what is central from what is peripheral. [End Page 7]
3. the divided line proportion
Understanding how Plato would have conceived an actual line divided in the proper proportions, with the mathematical knowledge and tools available to him, requires a return to the constructive methods of geometric proof that were in the process of being codified during his lifetime and that were compiled soon after in the form we know today as Euclid's Elements. Translating Plato's descriptions into a diagram that makes good on both the desiderata of what has seemed for centuries to be a contradiction reveals a complex relationship that would otherwise be invisible—and has been invisible. The topographical relations given by a diagram add another dimension for the representation of concepts that may seem ambiguous in everyday language.28
The construction of the divided line proportion (DLP) depends on notions developed in Euclid's Elements 2, which continues with the theme of transformation and application of areas with the now disputed concept of a geometric "algebra" that allowed for solutions to problems that contained the general form of quadratic equations.29 Five of the fourteen propositions begin with the accustomed phrase, "If a straight line be cut at random" (2.2–4, 7–8). Despite the lack of a general theory of proportion, the use of rectangular parallelograms and parallelogrammic areas makes possible additional diagrammatic proofs of equal areas and equal segments.
The DLP, using as a unit the smallest whole number, for simplicity, is the familiar 4 : 2 :: 2 : 1 (or the proportion p2 : pq :: pq : q2).30 It is closely related to the geometric mean or mean proportional; and the construction of a continuous geometrical proportion is central to demonstrating how Plato's divided line would have been drawn by those familiar with the geometric methods of the time.31 Since the construction of geometric means is the mathematical concept that justifies my diagram of Plato's divided line, it requires some elaboration here. [End Page 8]
The actual division of the line turns out to be straightforward, yet has a considerable amount of theory behind it because the proportion must apply to any line arbitrarily divided that could produce irrational quantities. Supplying a proof or calculating particular quantities to satisfy the ratios is much more complicated than the actual division. Dividing Plato's line could even seem anti-climactic to some, for it is as simple as applying ratios from one line to another using similar triangles involving parallel lines, and dropping perpendicular lines from a point to mark a length on another line. This construction, using the geometric mean, has the advantage over others that a right-angled triangle inherently sets up a series of parallels showing the replication of the ratios, following from the construction lines through the application of what is now called the triangle proportionality theorem.32
4. what glaucon drew
Like Greek mathematicians long before Euclid or even Socrates, and like some mathematicians today,33 Plato used diagrams (διάγραμμα) as proofs.34 The fact bears repeating because its importance has not been adequately appreciated: symbolic algebra has largely superseded geometry in modern mathematical thinking, but familiarity with some of the geometrical principles with which Plato was operating are essential for understanding Plato's text and, moreover, for seeing what Glaucon drew. I concede that it looks complicated at first glance, but that is because we spend little time in school with compasses. Facility is a matter of practice, and one quickly becomes adept.
I proffer the uncontroversial facts that (i) symbolic algebra was not available in the fifth and fourth centuries BCE, when (ii) diagrams were constructed with a compass and straightedge. But these were not today's compass and straightedge: the compass was collapsible, so the radius was lost after each inscription; and the straightedge was not a ruler for it had no unit marks. Moreover, as is widely repeated in the mathematics literature, Plato used only a compass.35 If so, let us not [End Page 9] be unduly surprised that drawing the divided line began with the circles required to determine its end points and establish its proportions.
It is difficult not to utilize the modern conceptions of abstract mathematical notions because this has been the language of discourse in which most of us have been introduced to them. Moreover, the geometric approach requires a visual graphic interface to enable us to think in this way, and it requires a step by step construction to actualize. The accompanying figures attempt to replicate that process, though I have omitted most of the repetitive steps of the construction because of their sheer number. Moreover, because omitting the role of the straightedge in a construction with so many circles would be almost impossible to apprehend, I include the triangles that are inscribed there. The core argument in favor of the DLP's being represented by a diagram that includes triangles is that the construction of a line divided in the proper proportions requires the auxiliary construction of triangles to carry out the division.36 There are several different, closely related, ways to perform this division; the one I offer reiterates the use of parallel lines for finding the mean proportional between two lengths.
The first figure in the sequence shows the construction of the geometric mean, which is the first step in the process: from the first cut, γ, a perpendicular line is drawn to connect to the circumference of a circle whose diameter is the original line being divided. As Elements 6.13 shows, this perpendicular line is the mean proportional to the two segments of the original line.37 The remaining figures illustrate the iterative process that would have been used to determine the successive geometric means by constructing rays (illustrated here with dashed lines) that project the original ratio onto a larger triangle, and back onto the line itself; that is, the auxiliary construction lines project the initial ratio onto the diagonal side of the triangle and back onto the line as the proportion p2 : pq :: pq : q2. In short, determining where Glaucon should make cuts δ and ε requires the series of constructions that include the triangle, shaded in the fourth figure.
The third figure in the sequence enables me to provide the geometric proof for the equality of the two middle segments of Plato's divided line that I promised in section 2. To see it in Plato's double sense of diagramma—diagram qua proof—consider the [End Page 10] third figure, but focus on the original line, the triangle, and the auxiliary parallel ray constructed immediately to the left of the divided line itself. The equality of the two middle segments would have been obvious to any Greek schoolboy looking at the diagram and understanding the meaning of 'parallelogram'—but it almost certainly requires a proof in words in our own time. Applying Elements 1.33, parallelograms between the same parallel lines, and with the same base, are equal in area. The original divided line and the ray are straight and parallel, so δζ, γη, and εθ are equal, forming the two parallelograms δζηγ and γηθε. According to Elements 1.34,38 since ζδ (ηγ) is equal to ηγ (θε), and δη (γθ) is common, the two sides ζδ (ηγ), δη (γθ) are equal to the two sides γη (θε), ηδ (θγ) respectively; and the angle ζδη (ηγθ) is equal to the angle δηγ (γθε); therefore the base ζη (ηθ) is also equal to γδ (εγ), and the triangle ζδη (ηγθ) is equal to the triangle γηδ (εθγ). Therefore, the diameter δη (γθ) bisects the parallelogram ζηγδ (ηθεγ), forming a third parallelogram δηθγ bisected by ηγ. Now we reach the clincher: since δγ and ηθ must be equal, and ηθ must be equal to γε, then the two middle segments of the divided line, δγ and γε, must be equal in length.39
My final illustration of the divided line, with most of the constructions removed but implicit, displays the DLP as a continuous geometric pro portion, able to depict the equality and the increase between the two middle sections simultaneously. The divided line—constructed as rays of an angle within a triangle, one ray [End Page 11] corresponding to cognition and the other to the objects of cognition—results in a divided line where increasing areas, not γε and δγ themselves, illustrate the increase in truth and clarity from Γ to Β. I resort to the term 'areas' instead of 'segments' for τμήματα—despite my note 4—to highlight what is made visible by the epistemological diagonal ray, and to distinguish it from the ontological vertical ray that has usually stood by itself as the divided line. My interpretation leaves one crucial construction line in view, an elongated reflection, as it were, of the vertical.
5. implications and puzzles
Some of the initial implications of what Glaucon drew have already been mentioned in passing: that no textual emendations are required, that Plato's knowledge of mathematics was more advanced than twentieth-century philosophers realized, and that the process of constructing the divided line is crucial to understanding it. Somewhat more can be said, however tentatively.
I agree with Foley that Plato is indicating "that the serious reader should analyze what these further difficulties might be"; but Plato is not "signaling that the core issue surrounding the divided line is the issue of contradiction," and challenging the reader to discover this contradiction.40 If, as I have argued, the τμήματα we see at 511d are segments of the above triangle, that is, areas rather than line segments, then the overdetermination problem disappears; there is no contradiction to be discovered. This leaves us with a quite different interpretation of 534a, where Socrates recommends to Glaucon that they not work out all of the ratios at this time, "to prevent its costing us many times the number of words we've used in discussing the preceding topics" (534a5–8). For Foley, Plato is here signaling that he is aware of the overdetermination problem.41 But if there is no overdetermination problem, then Plato's transposition of δγ and γε at 534a3–5 may be innocuous, a gesture to the equality of δγ and γε.
Even Plato's innocuous gestures, however, can raise devilish questions of interpretation. Why even mention a required lengthy discussion if anyone with a basic understanding of geometrical diagramming would recognize that δγ and γε are equal in length? I suggest that Plato is addressing not mathematicians but philosophers with his comment. One possibility is that descriptions in words of simple mathematical operations visible to the eye would be a distraction in the context of the conversation—much as I omitted repetitive steps in my illustration of the construction. Another possibility is that it is not to the illustrated ratios that Plato refers, but to the further ratios that would be required to generate the hierarchies of relationships among the entities each segment is said to represent. To put it another way, what if further divisions of each segment, by the same ratio, were necessary for complete clarity about the relations among entities? Consider for a moment those two middle segments and how they might be populated. If, for example, the being of a tree is different from the being of a shield, one might call for a division of γε, sensible objects, to account for the difference between natural [End Page 12] objects and artefacts. Since some natural objects are alive, some are not only the objects of sensation but subjects who themselves sense, which provides another joint where natural objects might be further divided. More controversial would be further divisions of δγ, mathematical objects—unless Plato would have thought it appropriate to lump together willy-nilly the number 4, the isosceles triangle, the even and the odd, formulae with pi, the hypotenuse, definitions, axioms, and propositions. Working out all the ratios is what would involve a lengthy discussion, detracting from the more general subject under discussion.
With Brumbaugh, I admitted earlier that Glaucon's line could not be linear and that it would illustrate visually the difference between what is central and what peripheral. Therein lies what I regard as an important implication of the passage as we have it, and of Glaucon's diagram: insofar as the vertical line is the explicit goal of the construction, ontology is prior to epistemology. Further, and according to the diagram, there are mathematical intermediates that are clearer and truer than sensible objects.42 Making the case that being is in fact prior to knowing, however, or that Plato consistently held that ontology is prior to epistemology, or even that propositions about the sphere itself are superior to—clearer and truer than—propositions about the cue ball in my hand, would make my paper "many times the number of words" it is already, for supporting evidence would need to be drawn not only from the text but from other dialogues, mostly from the late group.43 For now, my aim is more limited.
Complex and fundamental metaphysics is not settled by diagrams because, as is true of all diagrams since they are all images, they can never get us all the way to complete and genuine understanding. One might then wonder whether I have not circled right back to conclusions about the inadequacy of the divided line diagram that others have offered, namely, that it is deliberately flawed (Smith) or contradictory (Foley). I have not: the correctly drawn image illustrates what the dialogue's characters say without distortion. Yet, Plato is right to criticize the inadequacy of all images, in this case because the four levels do not illustrate all that we want to understand about being and knowing.
One is nevertheless entitled to frustration that Plato was not more direct about the relationships depicted in the diagram, that is, that the text does not provide more clues, more hints, about how its correct interpretation might move us further along toward the goal of understanding.44 At the very least, the [End Page 13] diagram I have offered is a visual aid to the resolution of both the ontological and epistemic aspects of the overdetermination problem, that is, the elimination of the seeming contradiction between 509d6–8 (with 534a3–5) and 511e2–4. It also provides a better understanding of how the mathematical practices of Plato's time may inform our interpretations of his texts. It is another matter whether Plato actually envisioned exactly this diagrammatic representation, but I have provided a justification that reflects accurately the concepts of Plato's text in a diagram that Glaucon could have drawn. It could plausibly have been conceived by Plato himself.45
Terry Echterling is Instructor of Philosophy at Michigan State University.
bibliography and abbreviations
1. In crucial respects, beginning here where the intelligible segment is longer than the visible segment, I regard Smith's arguments and description ("Divided Line," 42–43) as dispositive. He concludes, "Plato's divided line is a vertical line, divided unequally with the largest segment on top. These two segments represent the intelligible realm (at the top) and the visible realm (at the bottom)." Smith usefully canvasses scores of previous constructions and discussions of the divided line, from which I have learned much. He shows clearly that twenty-three then-recently published images of the line—including the well-known representations by Allan Bloom, Robert S. Brumbaugh, and G. M. A. Grube—do violence to the text of Plato's Republic. Popular illustrations of the line in more recent translations continue the injustice (Tom Griffith and G. R. F. Ferrari, C. D. C. Reeve, R. E. Allen, and Alain Badiou). Some commentators' diagrams so ill fit their own words that, so as not to be uncharitable, one is tempted to blame deficiencies in the history of printing and illustrating.
2. For the sake of exposition, I am using the term 'ratio' in the modern sense. The Greek notion of proportion (a is to b as c is to d) differed from their notion of ratio; and there were at least three competing definitions of 'ratio': from music theory, astronomy, and mathematics. See David H. Fowler, "Ratio" and A. Thorup, "Pre-Euclidean Proportions." Euclid makes no effort to reconcile the different senses of proportion given in books 5 and 7; and Plato's Philebus 25a introduces two senses as well, as pointed out by Benno Artmann ("Prehistory," 6).
3. Smith, "Divided Line," 42, and Foley "Undividable," 7. Plato, like Euclid, used 'segment/s' (τμῆμα, pl. τμήματα) broadly to denote not just parts of lines but parts of circles and polygons as well (e.g. Symposium 191d6). The divided line passage is replete with instances of its cognates but, as in English, the reader—or observer—must determine its referents in context. This point will become crucial when I introduce what Glaucon drew.
4. Republic translations are those of Christopher Rowe, and references to the Greek text are to the OCT edition of S. R. Slings—neither of whom includes a diagram of the divided line.
5. Rowe notes that "the language is that of the argument with the sight- and sound-lovers in 475d–480a."
6. In a masterful and comprehensive updating of the literature on the matter, Foley ("Undividable") concentrates on the incompatibility between 509d6–8 and 511e2–4 and how that incompatibility is intensified by 534a, carefully combing through forty-eight philosophical and philological attempts to address the problem.
7. Paul Pritchard (Plato's Philosophy of Mathematics, 92) argues that the objects of the two middle segments are the same, except that in the higher, "they are now being used as images of something else." Pritchard is not alone in making the claim; I mention him because he escaped the very wide nets of both Smith and Foley.
8. To take a simple example, one's imagined cue ball is less cognitively reliable than one's perceptions of the cue ball as it is held, felt, and seen. But the cue ball, as weighed and measured to the very limits of our most precise instrumentation, is but an imperfect image of the sphere itself, about which one could say in truth that its surface is 4 × × r2. Just how true is a further issue. In his discussion of Plato's divided line, Nicholas Rescher ("Epistemology," 154) notes that the highest kind of knowledge requires a scientific framework that is thus best captured by Spinoza's notion of adequacy. If s is an infinite series, one might question whether the formula is a complete truth.
9. For discussion of revisionists, see Foley, "Undividable," 8–9; demarcation advocates, 9–12; those claiming a blunder, 12–15; and dissolutionists, 15–17 (citing Smith, "Divided Line," 40n34). For clarity, here and below, I refer to the labels of the line I introduced above, rather than to labels other authors assigned.
10. Smith says, "I am tempted to think that Plato might have purposefully woven this subtle flaw into the intricate fabric of his own image, because he wished to avoid the sin of perfection. According to his own philosophy, images can never be perfect, and Plato's divided line is, after all, only an image" ("Divided Line," 43). Foley says, "Plato presents the divided line in a contradictory fashion because in so doing he forces the reader to follow the paradigmatic course in the procurement of true wisdom: from mathematics to philosophy ("Undividable," 23).
11. Exactly the same procedure would be available if one wanted different ratios, say, 30 : 6 :: 25 : 5 :: 5 : 1 to expand the length of the segment of intelligibles.
12. Foley ("Undividable," 21n43) draws attention to the connection between physically drawing Plato's line and recognizing that a problem exists: "I discovered the overdetermination problem drawing the line in front of a class. I simply could not get the middle two subsegments to look right, since I believed that one should be longer than the other, given what Plato says about the corresponding types of mental states." Nicholas Denyer ("Sun and Line," 292–93) contributes amusing visualizations of the line on behalf of Proclus and Plutarch, respectively, both of which generate the appearance of a 4–2–2–1 line, keeping the middle segments equal, and noting that Plutarch's version differs in making εἰκασία (fancy) the longest segment.
14. "Undividable," 8.
15. There has long been a stream of mathematicians who publish on Plato in books and journals that historians of ancient philosophy rarely encounter. Having studied art and ancient Greek mathematics (Wilbur Knorr, Evolution; and Árpád Szabó, Beginnings) on my way to Plato and philosophy, my "solution" to the mystery of Plato's divided line was originally submitted in an undergraduate term paper with a hand-drawn illustration that I still prefer to what I can produce with software.
16. Artmann ("A Proof," 18). See J. J. Coulton, "The Meaning of Ἀναγραφεύς"; L. Haselberger, "The Construction Plans for the Temple of Apollo at Didyma"; and John R. Senseney, The Art of Building in the Classical World.
17. Fowler, our most comprehensive source for mathematics as it was practiced in Plato's Academy, notes in passing that "the word diagramma seems, in Plato and Aristotle, to refer ambiguously to either a geometrical figure or a proof" (Mathematics, 33). Other philosophers have made similar, though less far-reaching, observations; for example, Richard Patterson ("Diagrams," 2) emphasizes the usefulness of diagrams in Plato's proofs. That diagrams are essential in geometry is more often pointed out. See, for example, Burnyeat, "Platonism," and Reviel Netz, "How Propositions Begin," 306n22, both cited by Hugh H. Benson, Clitophon's Challenge, 252n46.
18. The origins of Greek mathematics is a vast research field, but see Artmann, "Prehistory"; Knorr, Evolution, and "Pre-Euclidean"; Maurice Caveing, "Debate"; Szabó, Beginnings; Thomas Little Heath, History; Netz, Shaping of Deduction; and Sabetai Unguru, "Rewrite the History." Heath's notes to his translation of Euclid's Elements remain invaluable.
21. Commentary, 119.
23. R. Robson and Walter F. Cannon, Royal Society, 169 (misremembered by Rescher as senior Wrangler, "Epistemology," 152n46—an error well-spotted by an anonymous referee). Whewell was later to include the address, entitled "Of the Intellectual Powers According to Plato," as an appendix to Discovery (440–48), itself more about Plato than about any other single topic. For Whewell, it was "the study of the exact sciences in a comprehensive spirit" that had the power to make a person dialectical, and enable the ascent to first principles (438). He proposed that the highest level of the divided line must include the axioms of mathematics, though its definitions and theorems are properly relegated to the level of hypotheticals because "the Axioms of Arithmetic and Geometry belong to the Higher Faculty, which ascends to First Principles" (441n3). Rescher's paper on Plato's line, which discusses Proclus, Whewell, and Henry Sidgwick—eschewing twentieth-century Platonists—proposes that the mathematical level "may encompass symbolically mediated thought in general" ("Epistemology," 135).
24. Discovery, 444n6 (notation slightly modified).
25. "Epistemology," 152, 156–63. Continuing with the implications of Rescher's approach to the interesting parallels he draws among the sun, line, and cave is far from my present task. Another post-Foley paper should be mentioned in the same vein (informative and seminal, but outside my current task): Benson, "Plato's Philosophical Method in the Republic: the Divided Line (510b–511d)," superseded by ch. 9 of his Clitophon's Challenge.
26. The quotations above are from Plato's Mathematical Imagination, 91–92. Brumbaugh wrestled with the divided line problem throughout his life and published other contributions over decades, but his discussions of the early 1950s seem to me most perceptive.
27. "Plato's Divided Line," 533–34.
29. The notion that the ancient Greeks had a form of algebra that used geometric elements (lines, areas, angles) as "symbols" for formal reasoning rather than letters and operational symbols (=, –, + etc.) is widely held, especially among mathematicians working on foundational aspects of mathematics. Although this view has been vigorously contested by Unguru, "Rewrite the History," the dispute may well resolve into a matter of semantics that depends on one's conception of algebra. Greek geometric "algebra" was not as formally abstract as modern algebra in that it also utilized visual reasoning and concepts we now associate with spatial notions and aspects of physics, but supporters of the notion of a "geometric algebra" emphasize the functional role it played in the investigation of mathematical structures and thus its similarity to algebra in the more general, contemporary use of the term.
30. Any geometric sequence that is a continued proportion in which the consequent of each ratio is the antecedent of the next could be used. The DLP is deceptively simple and very uninformative, especially if stated in a modern algebraic form where letters designate each segment on the line: p2 / pq = pq / q2. It can be reduced to a continuous proportion that more clearly shows one aspect of the relationship but loses the four-part aspect: p2 : pq : q2. The geometric aspect can be shown by p2q2 = (pq)2 or by pq = p2q2.
31. In a sense, the DLP can be considered somewhat similar to the classic problems of duplicating the cube and squaring the circle, though the scope of the DLP is much broader. Its construction and proof involve the theory of proportions and irrational numbers that were foundational issues addressed adequately only when the first ten books of Euclid's Elements appeared; even they give only part of the ancient Greek geometrical thinking (structures behind harmonics and astronomy are not fully included). It is far beyond the scope of this paper to plot the connections between the DLP and propositions in Elements, or other passages in Plato that support my view of his grasp of the fundamental issues of mathematics.
32. This, like Meno 82b–85b, is an occasion when Plato gives an accessible demonstration of higher mathematics from more intuitive notions.
33. The chief proponent of diagrammatic proof is James Robert Brown, "Peeking into Plato's Heaven," and Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Denyer, while acknowledging the use of diagrams as proofs, cautions against becoming promiscuous about it ("Sun and Line," 294–303).
34. See Fowler, Mathematics, 33; Artmann "A Proof"; Knorr, "Construction as Existence Proof"; Ian Mueller, "Mathematical Method," especially 184–85; and Hieronymus Georg Zeuthen, "Geometrische Construction."
35. See "Geometric Construction" in Weisstein, Concise Encyclopedia of Mathematics, 1185–86, which adds, "It turns out that all constructions possible with a compass and straightedge can be done with a compass alone, as long as a line is considered constructed when its two endpoints are located." Thanks to David C. Royster for advice about early sources for this information.
36. Geometers will recall that Elements 1.1 uses circles to construct a triangle on a straight line. Advantages of a triangle for the ascent from the cave in Republic 7 (see Henry Jackson, "On Plato's Republic VI, 509d sqq."). Analogous "ascent" passages in Symposium and Phaedrus, as well as mathematical passages Meno—while interesting and important—are outside the scope of the present discussion.
37. See Elements 2.14 for an alternative proof for the correctness of the construction.
38. Only the variables differ from a verbatim quotation of Elements 1.34.
39. Derivatively, the corresponding segments on the diagonal ray that forms the triangle are equal in length.
40. "Undividable," 18–19.
41. "Undividable," 22.
42. What those intermediates are remains open to dispute: recall that Whewell counted the axioms of mathematics among the forms (Discovery, 441n3), in a sense dividing one of the line's "objects." The more common—though also disputed—approach is to hold the objects fixed while allowing more fluidity to cognition, that is, to permit both knowledge of sensibles and beliefs about forms. See Smith, "Divided Line," 34–35 with n24.
43. While this is not the place for a full discussion of the relationship between mathematical and sensible objects, I think Plato recognized that they are intertwined in a way that makes it difficult to understand one without the other. As we might now say, it is a relationship sharing in complexity the difficulties that are encountered when numerical concepts are applied to geometric magnitudes or the reverse.
44. Responses with which the literature is littered will be familiar: Plato saves his most complex views for initiates; Plato is only providing a backdrop for politics and moral psychology, so it is unreasonable to expect him to express his more fundamental positions in detail; Plato, at the time of writing the Republic, still had not settled all the facets of his mature view of mathematics; Plato would have needed further divisions of the divisions of the line to accomplish a comprehensive picture, and would thereby have detracted from the line's parallel to the cave. I have sided implicitly with the last.
45. Parts of this paper were presented to the Society for Ancient Greek Philosophy in 2008 and the Ancient Philosophy Circle at Michigan State University in 2013. I thank both audiences for their comments. I am especially grateful to Debra Nails and Emily Katz for the invaluable conversations and shared insights that greatly broadened my interpretation of the construction, and to the readers for the Journal of the History of Philosophy for their careful comments.